The problem of transient two-dimensional transport by diffusion and advection of a decaying contaminant in two adjacent porous media is solved using a boundary-integral method. The method requires the construction of appropriate Green's functions. Application of Green's theorem in the plane then yields representations for the contaminant concentration in both regions in terms of an integral of the initial concentration over the region's interior and integrals along the boundaries of known quantities and the unknown interfacial flux between the two adjacent media. This flux is given by a first-kind integral equation, which can be solved numerically by a discretisation technique. Examples of contaminant transport in fractured porous media systems are presented.

With the help of the generalized Mayer theorem we obtain an improved inequality for free energies of model and approximating systems, where only “connected parts” over the approximating Hamiltonian are taken into account. For a concrete system we discuss the problems of convergence of appropriate series of “connected parts ”.

The likelihood ratio approach to the detection of small signals in the presence of noise is investigated in the case where the available data have been clipped. The statistic obtained is the ratio of orthant probabilities and appears intractable; accordingly an approximation to this statistic is developed by truncating an appropriate Taylor expansion. Approximations are obtained for the mean and variance of this modified statistic and compared with those obtained from computer simulations.

The temporal instability of a developing swirling incompressible jet is considered. The jet development (in the streamwise direction) is modelled by combining a near-field and far-field approximation to the jet velocity profile into a one parameter family of basic velocity fields. The single parameter in the jet velocity field then allows us to model the radial spreading of the jet and the decay of swirl observed experimentally. Two distinct modes of instability of this model profile are found. The first is that found from a stability analysis of a fully developed swirling jet in the far field whilst the second is relevant to a “top-hat” jet with an imposed rigid body rotation. We demonstrate that the effect of azimuthal swirl is to destabilise both modes of instability. Additionally our results suggest that the near-nozzle modes of instability will dominate; indeed the growth rates of these modes are significantly larger than those found from previous studies of a fully developed jet in the far-field region.

Using geometric quantization, and accepting the quantum Hamiltonian of previous authors, we propose some candidate formulae for the quantum operator of an observable which is a quadratic form in the momenta.

Multiwavelets possess some nice features that uniwavelets do not. A consequence of this is that multiwavelets provide interesting applications in signal processing as well as in other fields. As is well known, there are perfect construction formulas for the orthogonal uniwavelet. However, a good formula with a similar structure for multiwavelets does not exist. In particular, there are no effective methods for the construction of multiwavelets with a dilation factor a (a ≥ 2, a ∈ Z). In this paper, a procedure for constructing compactly supported orthonormal multiscaling functions is first given. Based on the constructed multiscaling functions, we then propose a method of constructing multiwavelets, which is similar to that for constructing uniwavelets. In addition, a fast numerical algorithm for computing multiwavelets is given. Compared with traditional approaches, the algorithm is not only faster, but also computationally more efficient. In particular, the function values of several points are obtained simultaneously by using our algorithm once. Finally, we give three examples illustrating how to use our method to construct multiwavelets.

The paper examines a matrix equation given by Ziebur [6] for the growth of a population in which the birth-rate and death-rate are age-dependent. For convenience the population was sub-divided into four age groups, with the same birth-rate and death-rate for individuals in a particular group, and the matrix equation relates the numbers in each sub-division in consecutive years. This avoids delay terms and makes it easier to modify the growth equation but it is shown that the form suggested by Ziebur for the transition matrix leads to some difficulties.

A detailed discussion of Newtonian and general relativistic spherically symmetric dust solutions leads to the following suggested criteria for a singularity to be classified as a shell-cross: (1) All Jacobi fields have finite limits (in an orthonormal parallel propagated frame) as they approach the singularity. (2) The boundary region forms an essential C2 singularity which is C1 regular, that is it can be transformed away by a C1 coordinate transformation.

We consider a family of two-point quadrature formulae, using some Euler-type identities. A number of inequalities, for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or R-integrable functions, are proved.

The completion-time variance (CTV) and the waiting-time variance (WTV) are two performance measures which are commonly used in optimization of single-machine scheduling systems. This paper shows that when the number of jobs is large the two measures are nearly equivalent in a probabilistic environment.

Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.

Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].

In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

The first part of this paper starts with a brief discussion of some methods for solution of nonlinear equations which have interested the first author over the last twenty years or so. In the second part we discuss a recent research involvement, the success of which relies heavily on the numerical solution of nonlinear equation systems. We briefly describe path-following methods and then present an application to a simple steady-state reaction-diffusion equation arising in combustion theory. Results for some regular geometric shapes are shown and compared with those from an approximate method.

The observables of modular quantisation are studied from the point of view of locality. Such a study allows identification of possible Hamiltonians and also enables us to generalize the fundamental trilinear commutation relations of parafield theory. A comparison of modular field theory with a normal U(m) gauge theory, begun in an earlier publication, is completed with the conclusion that the two are equivalent except that the former has certain restrictions on its observables.

This paper considers similarity solutions of the multi-dimensional transport equation for the unsteady flow of two viscous incompressible fluids. We show that in plane, cylindrical and spherical geometries, the flow equation can be reduced to a weakly-coupled system of two first-order nonlinear ordinary differential equations. This occurs when the two phase diffusivity D(θ) satisfies (D/D′)′ = 1/α and the fractional flow function f (θ) satisfies df/dθ = kDn/2, where n is a geometry index (1, 2 or 3), α and k are constants and primes denote differentiation with respect to the water content θ. Solutions are obtained for time dependent flux boundary conditions. Unlike single-phase flow, for two-phase flow with n = 2 or 3, a saturated zone around the injection point will only occur provided the two conditions and f′(1) ≠ 0 are satisfied. The latter condition is important due to the prevalence of functional forms of f (θ) in oil/water flow literature having the property f′(1) = 0.

It has been known for some time that the Boltzmann weights of the chiral Potts model can be parametrised in terms of hyperelliptic functions. but as yet no such parametrisation has been applied to the partition and correlation functions. Here we show that for N = 3 the function S(tp) that occurs in the recent calculation of the order parameters can he expressed quite simply in terms of such a parametrisation.

We discuss Ablowitz-Segur's connection problem for the second Painlevé equation from the viewpoint of WKB analysis of Painlevé transcendents with a large parameter. The formula they first discovered is rederived from a suitable combination of connection formulas for the first Painlevé equation.